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1. Who may blow?

Men below 18: On the first day of Rosh Hashanah, a man cannot be Yotzei the blowing of Shofar with the blowing of another man unless that man is above age 18 or has grown a nice amount of facial hair. On the second day of Rosh Hashanah, a man may be Yotzei the blowing of Shofar with the blowing of a boy above Bar Mitzvah even if the boy is under age 18 and has not begun to grow facial hair. Nonetheless, initially, it is proper to be Yotzei the Shofar of even the second day through a man who is above age 18 or has grown facial hair. See Chapter 10 Halacha 3!

Women? Women may blow Shofar with a blessing on behalf of themselves or another woman.[1] However, they may not blow on behalf of another man[2] or even a male child.[3] Furthermore, it is best for a man to blow Shofar for a woman rather than have a woman blow for herself or for another woman.[4] Nevertheless, from the letter of the law she may do so.[5]

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[1] Admur 589:2; Michaber 589:6; R”H 33a as rules Rebbe Yossi and Rebbe Shimon; Rabbeinu Tam; Ran; Rosh there; Beis Yosef 589; See Kaf Hachaim 589:22

The reason it is allowed for women to blow Shofar on Rosh Hashanah despite it being a Shvus: Although blowing Shofar on Shabbos or Yom Tov is forbidden if it is not done for the sake of the Mitzvah and women are not obligated in the Mitzvah nevertheless the Sages allowed women to blow it if they intend to do so for the sake of the Mitzvah. The reason for this is because this specific prohibition of blowing a Shofar is not an actual complete Shvus as it is merely forbidden due to Uvdin Dechol, thus for the sake of appeasing the women the Sages permitted this light Rabbinical matter. [589:2]

[2] M”E 589:8; Kaf Hachaim 589:15

[3] Halichos Beisa 15 footnote 91 in name of Rav SZ”A; Piskeiy Teshuvos 589:2

[4] Shaagas Aryeh 105; Machazik Bracha 589:4; Shaareiy Teshuvah 589; Kaf Hachaim 589:26

[5] Piskeiy Teshuvos 589:2

[6] The reason: As seemingly, the entire law of verifying the growth of two hairs by a Biblical command is for the person to initially be Yotzei, if however, in truth the boy has two hairs, then even if this was unbeknownst to the person, he is Yotzei Bedieved, in the eyes of Heaven, as in truth the boy is an adult! Now, although it’s possible to learn that until testimony is given regarding his development, he is considered a child even in the eyes of heaven regarding another person, since it is customary for all Jewry to immerse in a Mikveh on Erev Rosh Hashanah, certainly testimony exists, and hence the man is Yotzei with him Bedieved. Accordingly, in Mivtzaim when dealing with an unobservant crowd who would not hear Shofar otherwise, one below age 18 who does not have facial hair should not be discouraged from blowing for other Jews, if they would not hear Shofar otherwise!